The scope of the LeChatelier Principle

LeChatelier [Comptes Rendus 99 (1884) 786; Ann. Mines 13 (2) (1888) 157] showed that a physical systemʹs “adjustment” to a disturbance to its equilibrium tended to be smaller as constraints were added to the adjustment process. Samuelson [Foundations of Economic Analysis, Harvard University Press, Cambridge, 1947] applied this result to economics in the context of the comparative statics of the actions of individual agents characterized as the solutions to optimization problems; and later (1960), extended the application of the Principle to a stable, multi-market equilibrium and the case of all commodities gross substitutes [e.g., L. Metzler, Stability of multiple markets: the hicks conditions. Econometrica 13 (1945) 277–292]. Refinements and alternative routes of derivation have appeared in the literature since then, e.g., Silberberg [The LeChatelier Principle as a corollary to a generalized envelope theorem, J. Econ. Theory 3 (1971) 146–155; A revision of comparative statics methodology in economics, or, how to do comparative statics on the back of an envelope, J. Econ. Theory 7 (1974) 159–172], Milgrom and Roberts [The LeChatelier Principle, Am. Econ. Rev. 86 (1996) 173–179], W. Suen, E. Silberberg, P. Tseng [The LeChatelier Principle: the long and the short of it, Econ. Theory 16 (2000) 471–476], and Chavas [A global analysis of constrained behavior: the LeChatelier Principle ‘in the large’, South. Econ. J. 72 (3) (2006) 627–644]. In this paper, we expand the scope of the Principle in various ways keyed to Samuelsonʹs proposed means of testing comparative statics results (optimization, stability, and qualitative analysis). In the optimization framework, we show that the converse LeChatelier Principle also can be found in constrained optimization problems and for not initially “conjugate” sensitivities. We then show how the Principle and its converse can be found through the qualitative analysis of any linear system. In these terms, the Principle and its converse also may be found in the same system at the same time with respect to the imposition of the same constraint. Based upon this, we expand the cases for which the Principle can be found based upon the stability hypothesis.